as3hx will convert only as3 code to haxe openfl and may well need hand tweaking. To make the code more flexible try to avoid extending Sprite then potentially it will be easier to use your activity logic in more webgl accelerated haxe solutions like Kha and Luxe.
You probably need clean seperation of the code and the onscreen layout.
To run a fla swf that is mostly graphics in openfl you need to use the haxelib https://github.com/openfl/swf, it can be quite fiddly getting to different children or frames and the gradients are not perfect, just remember when using it that it's not the same as accessing a child normally your using a structure that is a bit different to do that.
Alternatively you may find flump useful it can be used with openflump.
https://github.com/SavedByZero/openflump exports from flash IDE.
It converts lots of your graphics to png sequences in nested structures, it would probably be worth rebuilding textfields yourself.
Openfl js can be slow I suggest you do some tests before committing to a solution, since Kha or Luxe may provide you a more shader modern approach to graphics than a flash emulation api solution, but then all you assets largely need to be images.
I doubt if there is any ideal way but getting your logic in Haxe is really good idea because then you can do c++, c#, js etc.. The first step if you choose openfl is to perhaps to try just porting parts of your code to haxe and then use them as a swc in your current projects this will allow you to gradually check your conversion and fix an issues easily within a haxe flash approach prior to trying to do the same in html haxe. You may find the swf library is not ideal for html5 and that png's are the way to go, Animate now provides export png sequence which I have used in starling as3. Another approach to consider is the away3d plugin for Animate it can convert flash timeline into typescript away3d (2D ) that you can control with javascript code ( even haxe you may have got from as3hx if you remove some of the flash specific haxe and adjust it to work with the away3d 2d but I guess it could be very complex ).